Permutation groups
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Symmetry group of the square
The symmetries of a square are the permutations of the four symbols 1,2,3,4. [Diagram goes here - download the original to see it.] Permutation groups are represented by the symbol S, For example, S4 is the permutation group of 4 symbols. As the above example for the equilateral triangle shows, each permutation can be represented by a mapping diagram. Anticlockwise rotation of a square through As another example, rotation anticlockwise of a square through is equivalent to the mapping [Diagram goes here - download the original to see it.] [Diagram goes here - download the original to see it.] A more convenient notation is [Diagram goes here - download the original to see it.] A yet further way of representing a permutation is by observing, for example, here that 1è2è3è4è1 This means 1 maps to 2, 2 maps to 3, 3 maps to 4, 4 maps to 1. If we loop the 4 back to the 1 then this cycle can be simply written as [Diagram goes here - download the original to see it.] That is, besides indicating the following cycle 1è2è3è4 this symbol means that the 4 is followed by the 1 4è1 This is another convention for writing permutations
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Contents of
Permutation groups
1 Permutation groups
2 Symmetry groups are also permutation groups
3 Symmetry group of the square
4 Rotation through phi
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